# Isentropic process

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In thermodynamics, an isentropic process or isoentropic process (ισον = "equal" (Greek); εντροπία entropy = "disorder"(Greek)) is one in which, for purposes of engineering analysis, one may assume that the process takes place from initiation to completion without an increase or decrease in the entropy of the system, i.e., the entropy of the system remains constant.[1][2] It can be proven that any reversible adiabatic process is an isentropic process. A simple more common definition of isentropic would be one that produces "No change in entropy".

## Background

The second law of thermodynamics states that,

$TdS \ge \delta Q$

where $\delta Q$ is the amount of energy the system gains by heating, $T$ is the temperature of the system, and $dS$ is the change in entropy. The equal sign will hold for a reversible process. For a reversible isentropic process, there is no transfer of heat energy and therefore the process is also adiabatic. For an irreversible process, the entropy will increase. Hence removal of heat from the system (cooling) is necessary to maintain a constant entropy for an irreversible process in order to make it isentropic. Thus an irreversible isentropic process is not adiabatic.

For reversible processes, an isentropic transformation is carried out by thermally "insulating" the system from its surroundings. Temperature is the thermodynamic conjugate variable to entropy, thus the conjugate process would be an isothermal process in which the system is thermally "connected" to a constant-temperature heat bath.

## Isentropic flow

An isentropic flow is a flow that is both adiabatic and reversible. That is, no heat is added to the flow, and no energy transformations occur due to friction or dissipative effects. For an isentropic flow of a perfect gas, several relations can be derived to define the pressure, density and temperature along a streamline.

Note that energy can be exchanged with the flow in an isentropic transformation, as long as it doesn't happen as heat exchange. An example of such an exchange would be an isentropic expansion or compression that entails work done on or by the flow.

### Derivation of the isentropic relations

For a closed system, the total change in energy of a system is the sum of the work done and the heat added,

$dU = \delta W + \delta Q\,\!$

The reversible work done on a system by changing the volume is,

$dW = -pdV\,\!$

where $p$ is the pressure and $V$ is the volume. The change in enthalpy ($H = U + pV\,\!$) is given by,

$dH = dU + pdV + Vdp\,\!$

Then for a process that is both reversible and adiabatic (i.e. no heat transfer occurs), $\delta Q_{rev} = 0\,\!$, and so $dS=\delta Q_{rev}/T = 0\,\!$. All reversible adiabatic processes are isentropic. This leads to two important observations,

$dU = \delta W + \delta Q = -pdV + 0\,\!$ , and
$dH = \delta W + \delta Q + pdV + Vdp = -pdV + 0 + pdV + Vdp = Vdp\,\!$

Next, a great deal can be computed for isentropic processes of an ideal gas. For any transformation of an ideal gas, it is always true that

$dU = nC_vdT\,\!$, and $dH = nC_pdT\,\!$.

Using the general results derived above for $dU$ and $dH$, then

$dU = nC_vdT = -pdV\,\!$, and
$dH = nC_pdT = Vdp\,\!$.

So for an ideal gas, the heat capacity ratio can be written as,

$\gamma = \frac{C_p}{C_V} = -\frac{dp/p}{dV/V}\,\!$

For an ideal gas $\gamma\,\!$ is constant. Hence on integrating the above equation, assuming a perfect gas, we get

$pV^{\gamma} = \mbox{constant} \,$ i.e.
$\frac{p_2}{p_1} = \left(\frac{V_1}{V_2}\right)^{\gamma}$

Using the equation of state for an ideal gas, $p V = n R T\,\!$,

$TV^{\gamma-1} = \mbox{constant} \,$
$\frac{p^{\gamma -1}}{T^{\gamma}} = \mbox{constant}$

also, for constant $C_p = C_v + R$ (per mole),

$\frac{V}{T} = \frac{nR}{p}$ and $p = \frac{nRT}{V}$
$S_2-S_1 = nC_p \ln\left(\frac{T_2}{T_1}\right) - nR\ln\left(\frac{p_2}{p_1}\right)$
$\frac{S_2-S_1}{n} = C_p \ln\left(\frac{T_2}{T_1}\right) - R\ln\left(\frac{T_2 V_1}{T_1 V_2}\right ) = C_v\ln\left(\frac{T_2}{T_1}\right)+ R \ln\left(\frac{V_2}{V_1}\right)$

Thus for isentropic processes with an ideal gas,

$T_2 = T_1\left(\frac{V_1}{V_2}\right)^{(R/C_v)}$ or $V_2 = V_1\left(\frac{T_1}{T_2}\right)^{(C_v/R)}$

### Table of isentropic relations for an ideal gas

 $\Bigg\lbrack \frac{T_2}{T_1} \Bigg\rbrack$ $=\,\!$ $\left (\frac{p_2}{p_1} \right )^\frac {\gamma-1}{\gamma}$ $=\,\!$ $\left (\frac{V_1}{V_2} \right )^{(\gamma-1)}$ $=\,\!$ $\left (\frac{\rho_2}{\rho_1} \right )^{(\gamma - 1)}$ $\left (\frac{T_2}{T_1} \right )^\frac {\gamma}{\gamma-1}$ $=\,\!$ $\Bigg\lbrack \frac {p_2} {p_1} \Bigg\rbrack$ $=\,\!$ $\left (\frac{V_1}{V_2} \right )^{\gamma}$ $=\,\!$ $\left (\frac{\rho_2}{\rho_1} \right )^{\gamma}$ $\left (\frac{T_1}{T_2} \right )^\frac {1}{\gamma-1}$ $=\,\!$ $\left (\frac{p_1}{p_2} \right )^\frac {1}{\gamma}$ $=\,\!$ $\Bigg\lbrack \frac{V_2}{V_1} \Bigg\rbrack$ $=\,\!$ $\frac{\rho_1}{\rho_2}$ $\left (\frac{T_2}{T_1} \right )^\frac {1}{\gamma-1}$ $=\,\!$ $\left (\frac{p_2}{p_1} \right )^\frac {1}{\gamma}$ $=\,\!$ $\frac{V_1}{V_2}$ $=\,\!$ $\Bigg\lbrack \frac{\rho_2}{\rho_1} \Bigg\rbrack$

Derived from:

$pV^{\gamma} = \text{constant} \,\!$
$pV = m R_s T \,\!$
$p = \rho R_s T\,\! \,\!$
Where:
• $p\,\!$ = Pressure
• $V\,\!$ = Volume
• $\gamma\,\!$ = Ratio of specific heats = $C_p/C_v\,\!$
• $T\,\!$ = Temperature
• $m\,\!$ = Mass
• $R_s\,\!$ = Gas constant for the specific gas = $R/M\,\!$
• $R\,\!$ = Universal gas constant
• $M\,\!$ = Molecular weight of the specific gas
• $\rho\,\!$ = Density
• $C_p\,\!$ = Specific heat at constant pressure
• $C_v\,\!$ = Specific heat at constant volume

## References

• Van Wylen, G.J. and Sonntag, R.E. (1965), Fundamentals of Classical Thermodynamics, John Wiley & Sons, Inc., New York. Library of Congress Catalog Card Number: 65-19470

### Notes

1. ^ Van Wylen, G.J. and Sonntag, R.E., Fundamentals of Classical Thermodynamics, Section 7.4
2. ^ Massey, B.S. (1970), Mechanics of Fluids, Section 12.2 (2nd edition) Van Nostrand Reinhold Company, London. Library of Congress Catalog Card Number: 67-25005